Flips and Terminations

该奖项支持Shokurov教授和Kachi教授的一个项目。本课题主要研究高维日志最小模型程序（LMMP）中的问题，主要研究4维及以上维度日志翻转的存在性问题。目的是从LMMP本身和一些低维的末端归纳得到这样的翻转。特别地，我们的目标是构造日志翻转，甚至完成第4维的lmpp。此外，还将给出家庭翻转的几何描述。Shokurov和Kachi还将开始研究所需的终止，并演示lmpp的进一步应用。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源中，它处理的图形可以在平面上由最简单的方程（即多项式）定义，或者在三维空间中由最简单的几何结构（如圆锥截面）给出。传统上，该领域与数学的大多数分支相互作用。如今，该领域不仅使用代数的方法，还使用分析、拓扑和数学物理的方法，相反，它在这些领域以及数论、物理学、理论计算机科学和机器人技术中也得到了应用。

9800807 Shokurov This award support a project of Professors Shokurov and Kachi. The project is focused on problems in the Log Minimal Model Program (LMMP) in higher dimensions, mainly, on the existence of the log flips in dimension 4 and higher. The aim is to obtain such flips inductively from the LMMP itself and some terminations in lower dimensions. In particular, the aim is to construct the log flips and even to complete the LMMP in dimension 4. In addition, a geometric description of flips in families will be given. Shokurov and Kachi will also start to investigate needed terminations and to demonstrate further applications of the LMMP. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials, or be given in the 3-space by the simplest geometric constructions, e. g., conic sections. Traditionally, the field interacts with most of branches of mathematics. Nowadays the field makes use of methods not only from algebra, but from analysis, topology and mathematical physics, and conversely is finding application in those fields as well as in number theory, physics, theoretical computer science, and robotics.